Cotangent - Math Open Reference

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In a right triangle, the cotangent of an angle is the length of the adjacent side divided by the length of the opposite side. In a formula, it is abbreviated to just 'cot'.

Of the six possible trigonometric functions, cotangent, secant, and cosecant, are rarely used. In fact, most calculators have no button for them, and software function libraries do not include them.

They can be easily replaced with derivations of the more common three: sin, cos and tan. Cotangent can be derived in two ways:

The inverse cotangent function - arccot

For every trigonometry function such as cot, there is an inverse function that works in reverse. These inverse functions have the same name but with 'arc' in front. So the inverse of cot is arccot etc. When we see "arccot A", we interpret it as "the angle whose cotangent is A".

cot 60 = 0.577 Means: The cotangent of 60 degrees is 0.577
arccot 0.577 = 60 Means: The angle whose cotangent is 0.577 is 60 degrees.

Sometimes written as acot or cot-1

Large and negative angles

In a right triangle, the two variable angles are always less than 90° (See Interior angles of a triangle). But we can in fact find the cotangent of any angle, no matter how large, and also the cotangent of negative angles. For more on this see Functions of large and negative angles.

Graph of the cotangent function

Because the cotangent function is the reciprocal of the tangent function, it goes to infinity whenever the tan function is zero and vice versa.

The derivative of cot(x)

In calculus, the derivative of cot(x) is –csc2(x). This means that at any value of x, the rate of change or slope of cot(x) is –csc2(x). For more on this see Derivatives of trigonometric functions together with the derivatives of other trig functions. See also the Calculus Table of Contents.

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